Persymmetric matrix

Last updated March 09, 2024

In mathematics, persymmetric matrix may refer to:

a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or
a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.

The first definition is the most common in the recent literature. The designation "Hankel matrix" is often used for matrices satisfying the property in the second definition.

Definition 1

Let $A = (a ij)$ be an $n \times n$ matrix. The first definition of persymmetric requires that

a_{ij}=a_{n-j+1,\,n-i+1}

for all $i, j$ .^[1]

For example, 5 × 5 persymmetric matrices are of the form

A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\a_{21}&a_{22}&a_{23}&a_{24}&a_{14}\\a_{31}&a_{32}&a_{33}&a_{23}&a_{13}\\a_{41}&a_{42}&a_{32}&a_{22}&a_{12}\\a_{51}&a_{41}&a_{31}&a_{21}&a_{11}\end{bmatrix}}.

This can be equivalently expressed as $AJ = JA T$ where $J$ is the exchange matrix.

A third way to express this is seen by post-multiplying $AJ = JA T$ with $J$ on both sides, showing that $A T$ rotated 180 degrees is identical to $A$ :

A=JA^{\mathsf {T}}J.

A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes called bisymmetric matrices.

Definition 2

The second definition is due to Thomas Muir.^[2] It says that the square matrix A = (a_ij) is persymmetric if a_ij depends only on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form

A={\begin{bmatrix}r_{1}&r_{2}&r_{3}&\cdots &r_{n}\\r_{2}&r_{3}&r_{4}&\cdots &r_{n+1}\\r_{3}&r_{4}&r_{5}&\cdots &r_{n+2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n}&r_{n+1}&r_{n+2}&\cdots &r_{2n-1}\end{bmatrix}}.

A persymmetric determinant is the determinant of a persymmetric matrix.^[2]

A matrix for which the values on each line parallel to the main diagonal are constant is called a Toeplitz matrix.

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<span class="mw-page-title-main">Matrix (mathematics)</span> Array of numbers

In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.

References

↑ Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-8018-5414-9 . See page 193.
1 2 Muir, Thomas (1960), Treatise on the Theory of Determinants, Dover Press, p. 419

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[1] Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins, ISBN 978-0-8018-5414-9 . See page 193.

[muir-2] 1 2 Muir, Thomas (1960), Treatise on the Theory of Determinants, Dover Press, p. 419

[1]

[2]

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematicsportal List of matrices Category:Matrices

Persymmetric matrix

Contents

Definition 1

Definition 2

See also

Related Research Articles

References